Two-step Perturbation Technique Research Articles

Thermo-Electro-Mechanical Analysis of Micropolar FGP Cylindrical Shell Covered with Piezoelectric Actuator Layers

An investigation is performed in this paper to analyze the nonlinear thermo-electro-mechanical response of long sandwich cylindrical shells with functionally graded porous (FGP) core and thin piezoelectric actuator layers. The FGP core of the sandwich shell is assumed to be temperature- and microstructure-dependent. It is assumed that the sandwich shell is subjected to uniform lateral pressure loading in a thermo-electrical environment. The equilibrium equations of the shell with infinite length are established with the aid of the virtual displacement principle and von Kármán kinematic assumptions. The governing equations are obtained in terms of displacement components based on the first-order shear deformation model of shallow cylindrical shells and the modified couple stress theory. These nonlinear differential equations are analytically solved for a sandwich shell having both the simply-supported and clamped–clamped edge conditions by employing a two-step perturbation technique. Analytical closed-form solutions are determined as a relationship including the load parameter and the mid-span deflection. The comparison examples are made with the existing results in the literature for a simple functionally graded shell, where good agreement is obtained. It is shown that the nonlinear response of the sandwich shell is highly dependent upon the temperature variation, power law index, porosity coefficient, couple stress parameter, piezoelectric layers, and geometrical parameters of the shell.

Thermal deflection and snap-buckling analysis of temperature-dependent FGP curved nanotubes via a nonlocal strain gradient theory

Present study deals with the nonlinear analysis of thermally induced deflection and snap-through buckling in functionally graded porous (FGP) curved nanotubes. It is assumed that the curved nanotube is rested on a two-parameter nonlinear elastic foundation. The curved nanotube is subjected to uniformly distributed transverse pressure loading and also uniform temperature rise. Distribution of thermomechanical properties in the thickness of curved nanotubes is radially graded according to a power law function. The even type of porosity distribution patterns is included into the formulation and all properties are considered as temperature dependent. The nonlocal strain gradient and uncoupled thermoelasticity theories are provided to analyze the geometrically nonlinear behaviors of curved nanotubes. The equilibrium equations are obtained based on the high-order shear deformation theory. To capture the large deformations, the von Kármán type of nonlinear kinematic assumptions is included. Two types of immovable boundary conditions are considered which are simply-supported and clamped-clamped. The equilibrium equations are reformulated and transferred into the dimensionless presentation. The closed-form expressions are provided to trace the thermal/mechanical load-deflection paths of curved nanotubes using the two-step perturbation technique. Numerical results of this study are compared with the available data in the literature for the macro-scale curved tubes. After that novel parametric results are provided to discuss the effects of nonlocal and length scale parameters, boundary conditions, porosity coefficient, foundation stiffness, power law exponent, and geometrical parameters.

An improved model to simulate the effective material properties of functionally graded porous plates

This paper develops an improved model to estimate the effective material properties of functionally graded porous plates with evenly and unevenly distributed pores. Based on the higher-order shear deformation plate theory and extended von Kármán-type equations, dynamic equilibrium equations of the plate were built with the effect of the viscoelastic foundation. By using a two-step perturbation technique, the nonlinear frequency and dynamic response were obtain. The effects of the pore volume fraction, porosity distributions, and ceramic mass fraction were investigated.

New insights into nonlinear stability of imperfect nanocomposite beams resting on a nonlinear medium

An analysis is performed in this research to study the nonlinear thermal stability of graphene platelet reinforced composite (GPLRC) beams based on the third-order shear deformation model of Reddy and von-Kármán kinematic assumptions. The influences of the three-parameter nonlinear hardening/softening elastic foundation and initial imperfection are also taken into account. GPLs are distributed in the layers of the composite media where the volume fraction of GPLs may be different in each layer, so a piecewise functionally graded media is achieved. The elasticity modulus of the composite media is estimated via the Halpin–Tsai rule which includes the dimensions of reinforcements while the thermal expansion coefficient and Poisson’s ratio are obtained via the simple Voigt’s rule. Three nonlinear and coupled governing equations are established using the concept of static version of the Hamilton principle. Due to the immovability of the edge supports, the equations are reformulated to reach two coupled equations in terms of lateral displacement and cross section rotation. For the case of a beam with both ends simply-supported, a two-step perturbation technique is implemented to extract the closed-form expression which provides the mid-span deflection as a function of temperature elevation. Results of this study are compared with the available data in the open literature with respect to simple cases. After that novel results are given to explore the effects of foundation parameters, beam geometry, GPL weight fraction and GPL patterns. It is shown that for sufficiently soft elastic foundation limit load type of instability occurs and the structure becomes imperfection sensitive.

Thermal Buckling and Postbuckling Behaviors of Couple Stress and Surface Energy-Enriched FG-CNTR Nanobeams

Small-sized structural elements such as beams, plates, and shells are usually used as nanomechanical resonators, nanoscale mass sensors, nanoelectromechanical actuators, and nanoenergy harvesters. At the nanoscale, the structures usually possess a high surface area-to-bulk volume ratio, leading to the free energy related to surface atoms becoming considerable compared to that of the bulk part. Earlier reports indicated several physical reasons for size-dependent phenomena, e.g., nonlocal stress, surface energy, and couple stress. To provide an in-depth insight into the mechanical behavior of small-scale structures, size-dependent continuum models including two or more physical factors have attracted the attention of the academic community. This research analyzes the thermal buckling and postbuckling characteristics of functionally graded carbon nanotube-reinforced (FG-CNTR) nanobeams with a tri-parameter, nonlinear elastic foundation and subjected to a uniform temperature rise. Chen-Yao’s surface energy theory and Yang’s symmetrical couple stress theory are combined to capture two types of size effects in nanobeams. The postbuckling model is formulated based on the Euler–Bernoulli deformation hypothesis and Euler–Lagrange equation. Using a two-step perturbation technique, the related postbuckling equilibrium path is determined. In numerical analysis, the impacts of surface energy, couple stress, elastic foundation, boundary conditions, geometric factor, layout type, and volume fraction of CNTs on the thermal buckling and postbuckling behaviors of nanobeams are revealed. It is indicated that considering couple stress or surface energy can lead to a significant increase in the postbuckling stability of nanobeams compared to the case in which it is not considered. In addition, there is a reverse competition between couple stress or surface energy effects on the thermal buckling responses of nanobeams. As the temperature rise will cause the material elastic moduli softening, the thermal buckling load–deflection curves of nanobeams with the temperature-independent case are much higher than those with the temperature-dependent cases.

Nonlinear Vibration Analysis of Functionally Graded Porous Plates Reinforced by Graphene Platelets on Nonlinear Elastic Foundations

This paper presents a nonlinear vibration analysis of functionally graded graphene platelet (GPL) reinforced plates on nonlinear elastic foundations. Uniformly or non-uniformly distributed internal pores were present in the plates. Based on the modified Halpin-Tsai micromechanics model and the extended rule of mixture, the material properties were evaluated. The governing equations, coupled with the effect of nonlinear foundations, were derived by using the higher-order plate theory and general von Kármán-type equations. A two-step perturbation technique was employed to obtain the nonlinear frequency and transient response. After the present method was verified, the effects of pores, GPLs, and elastic foundations were investigated in detail. A new finding is that the influence of the porosity coefficient on the natural frequency and dynamic response is relevant to foundation parameters. Moreover, the influence of the nonlinear foundation parameter can be negligible.

Thermomechanical analysis of snap-buckling phenomenon in long FG-CNTRC cylindrical panels resting on nonlinear elastic foundation

Snap-buckling phenomenon in an infinitely long nanocomposite cylindrical panel subjected to uniformly distributed transverse pressure loading is analyzed in this research. It is assumed that the functionally graded (FG) carbon nanotube reinforced composite (CNTRC) shell is resting on nonlinear elastic foundation. The uniform thermal field is also included into the formulation. Thermomechanical properties of the shell are assumed to be temperature dependent. The CNTs distribution pattern can be uniform or functionally graded through the shell thickness. The shell is formulated using a general high-order shear deformation theory. Governing equilibrium equations of the shell which undergoes thermal–mechanical coupling loading are established via the virtual displacement principle. To capture the large deflections, the von Kármán type of kinematic assumptions is utilized. Two types of edge conditions are considered for the shell with infinite length which are immovable simply supported and clamped–clamped. To obtain the limit buckling load of the shell, the two-step perturbation technique and Galerkin procedure are used. Results of this study reveal that the snap-buckling phenomenon can take place in a shallow cylindrical panel subjected to lateral pressure loading. This attractive behavior is observed due to the immovability of the flat edges of the shell. It is shown that the snap-buckling phenomenon in a long cylindrical panel is highly dependent upon the elastic foundations, thermal environments, geometrical parameters and material properties.

Nonlinear thermal instability and vibration analysis of pre/post-buckled FG porous nanotubes using nonlocal strain gradient theory

Current research presents the nonlinear static/dynamic response of functionally graded (FG) porous nanotubes in the thermal pre- and post-buckling regimes. The case of temperature-dependent FG nanotubes with even distributed porosities subjected to uniform temperature rise is considered. The stability and vibration characteristics are analyzed for the uniformly heated nanotube surrounded by a nonlinear elastic foundation. The nonlinear equations of motion are derived using the Hamilton principle for the thermally pre/post-buckled nanotube in the framework of nonlocal strain gradient theory. The size-dependent governing equations are established based on the uncoupled thermoelasticity, the von Kármán assumption and the high-order shear deformation theory. The nonlinear governing equations are reduced for the nonlocal strain gradient-based model of nanotubes with immovable simply supported boundary conditions. The system of equations is solved for the nonlinear problems of free vibration and post-buckling of the heated tube using two-step perturbation technique and Galerkin procedure. The novel parametric examples are carried out to investigate the effect of nonlocal and length scale parameters, porosity distribution, functionally graded pattern, elastic foundations and geometrical parameters on the thermoelastic response of tubes. The obtained results highlight the importance of static/dynamic analysis of FG porous nanotubes in the thermal pre-buckling and post-buckling regimes.

Perturbation Method for Thermal Post-Buckling Analysis of Shear Deformable FG-CNTRC Beams with Different Boundary Conditions

This research aims to analyze the thermal buckling and post-buckling of carbon nanotube (CNT) reinforced composite beams. It is assumed that the beam is rested on a nonlinear elastic foundation which contains the Winkler spring, shear layer, and nonlinear spring. Distribution of CNTs across the thickness may be non-uniform which results in a functionally graded media. The elastic properties of the beam are evaluated using the refined rule of mixtures which contains efficiency parameters. Temperature dependency of the constituents is also taken into account. Using three different beam models, namely, first-order, third-order, and sinusoidal theories, the governing equations for the composite beam are established. Three different types of edge supports are considered which are pinned–pinned, clamped–clamped, and clamped–roller. With the aid of the two-step perturbation technique, closed-form expressions are extracted to obtain the elevated temperature as a function of the post-buckling deflection in the beam. Results of this study are compared with the available data in the literature. After that, new results are given to discuss the effects of important factors such as foundation parameters, geometrical characteristics, boundary conditions, the CNT volume fraction, and CNT pattern. It is shown that the critical buckling temperature of pinned–pinned and clamped–roller beams is the same while their post-buckling responses are totally different.

Free vibration and snap-through instability of FG-CNTRC shallow arches supported on nonlinear elastic foundation

This paper analytically investigates the nonlinear free vibration and snap-through instability of nanocomposite shallow arches reinforced with carbon nanotubes (CNTs). The functionally graded (FG) carbon nanotube reinforced composite (CNTRC) arch with shallow curvature is analysed under uniformly distributed transverse loading. The arch is assumed to rest on a three-parameter nonlinear elastic foundation in a uniform temperature field. Thermo-mechanical properties of the arch are graded through the thickness and are considered to be temperature-dependent. The equations of motion are established based on a general high-order shear deformation theory. These nonlinear equations are derived by Hamilton’s principle within the framework of the von Kármán assumption. The static equilibrium equations and dynamic equations of motion are solved analytically for the arch with immovable simply supported edges. The two-step perturbation technique and Galerkin method are implemented to obtain the closed-form solutions. The novel results illustrate the influences of CNT distribution pattern, foundation stiffness and geometrical parameters on the linear/nonlinear frequency and snap-through instability of the arch. It is shown that the minimum value of limit loads/frequencies belongs to the FG-O type of nanocomposite arches and the maximum ones are obtained in FG-X pattern.

Nonlinear size-dependent bending and forced vibration of internal flow-inducing pre- and post-buckled FG nanotubes

Fluid-conveying nanotubes are important components in nano-electromechanical systems requiring strict static and dynamic stiffness designs. The nonlinear bending and forced vibration of the fluid-conveying functionally graded nanotubes in pre- and post-buckling states are investigated. Based on Zhang–Fu’s refined displacement field, a comprehensive size-dependent nanotube model is established in which the nonlocal stress, strain gradient and surface energy effect are coupled in the constructive model. The nanotube is assumed simply supported at both ends and subjected to transverse uniform static or harmonic load. The two-step perturbation technique is extended to obtain the post-buckling equilibrium paths, providing the initial configurations for bending and resonance analysis. According to the two symmetric initial bifurcation paths, the bending load–deflection explicit relation is obtained by the two-step perturbation method again. A combination of two-step perturbation and modified Lindstedt–Poincaré method is developed to obtain the approximate analytical solution of the strongly nonlinear forced vibration. Parameter investigations are conducted to discuss the different size-dependent effects, flow velocity and material distributions on the nonlinear behaviors. Results reveal the dual influences of structural size-dependent effects and indicate that the response is more sensitive to the size-dependent effects when the flow velocity is close to the critical buckling velocity.

Large deflection analysis of FG-CNT reinforced composite pipes under thermal-mechanical coupling loading

This study is concerned with the analysis of large deflection in composite pipes reinforced with carbon nanotubes (CNTs) resting on nonlinear elastic foundation. It is assumed that the nanocomposite pipe is subjected to uniformly distributed transverse pressure loading and also uniform temperature rise. Distribution of CNTs is functionally graded (FG) through the radius of composite pipe and material properties are assumed as temperature-dependent. Two kinds of immovable boundary conditions are considered for the composite pipe which are simply-supported and clamped. The principle of virtual displacement is employed to establish the nonlinear equilibrium equations of the composite pipe under thermo-mechanical loading. Governing equations are extracted based on the higher-order shear deformation theory and the von Kármán geometric measure. The system of nonlinear differential equations is analytically solved utilizing the two-step perturbation technique and Galerkin method. Results of this paper are compared with the available data in the literature for the homogeneous isotropic pipes. The effects of boundary conditions, geometric factors, CNTs distribution profile, foundation stiffness, and the volume fraction coefficient on the bending behavior of FG-CNT reinforced composite pipes are discussed in detail. The novel results of this study demonstrate that the maximum deflection of the nanocomposite pipe occurs in the FG-Λ pattern and the minimum deflection belongs to the FG-V case.

Vibrational behavior of thermally pre-/post-buckled FG-CNTRC beams on a nonlinear elastic foundation: a two-step perturbation technique

Vibrational behavior of thermally pre-/post-buckled FG-CNTRC beams on a nonlinear elastic foundation: a two-step perturbation technique

Nonlinear analysis of size-dependent frequencies in porous FG curved nanotubes based on nonlocal strain gradient theory

A nonlocal strain gradient model is developed in this research to analyse the nonlinear frequencies of functionally graded porous curved nanotubes. It is assumed that the curved nanotube is in contact with a two-parameter nonlinear elastic foundation and is also subjected to the uniform temperature rise. The non-classical theory presented for curved nanotubes contains a nonlocal parameter and a material length scale parameter which can capture the size effect. A power law distribution function is used to describe the graded properties through the thickness direction of curved nanotubes. The even dispersion pattern is used to model the porosities distribution. The high-order shear deformation theory and the von Karman type of geometric non-linearity are utilized to obtain the nonlinear governing equations of the structure. The size-dependent equations of motion for the large amplitude vibrations of curved nanotubes are obtained by employing Hamilton’s principle. The analytical solutions are extracted for the curved nanotube with immovable hinged-hinged boundary conditions. Size-dependent frequencies of the curved nanotube exposed to thermal field are obtained using the two-step perturbation technique and Galerkin procedure. The effects of important parameters such as nonlocal and length scale parameters, temperature field, elastic foundation, porosity, power law index and geometrical parameters are studied in detail.

Thermally induced nonlinear stability and imperfection sensitivity of temperature- and size-dependent FG porous micro-tubes

An attempt is made in the current research to analyse the nonlinear thermal stability and imperfection sensitivity of functionally graded (FG) porous micro-tubes. Temperature-dependent properties of the geometrically imperfect micro-tube are graded across the radius of cross-section. It is assumed that the micro-tube with different end conditions is in contact with a two-parameter elastic foundation. The nonlinear component of the elastic foundation can be of the hardening or softening type. The equilibrium equations are obtained within the framework of von Karman nonlinear assumptions and high-order shear deformation tube theory. The governing equations are reformulated for the case of imperfect micro-tubes based on the modified couple stress theory. The system of nonlinear differential equations is solved using the two-step perturbation technique and Galerkin procedure. The analytical solutions are obtained for three different types of immovable boundary conditions which are clamped-rolling, simply-supported and clamped–clamped. The closed-form expressions are given to obtain the large deflection in the micro-tube as a function of the elevated temperature. Novel parametric studies are given to explore the thermal stability and imperfection sensitivity analysis of the perfect and imperfect micro-tubes, respectively. The effects of boundary conditions, couple stress components, porosity coefficient, elastic foundation, FG pattern, temperature dependence and geometrical parameters are studied.

On snap-buckling of FG-CNTR curved nanobeams considering surface effects

The aim of this paper is to analyze the nonlinear bending of functionally graded (FG) curved nanobeams reinforced by carbon nanotubes (CNTs) in thermal environment. Chen-Yao' s surface elastic theory and geometric nonlinearity are also considered. The nanobeams are subjected to uniform loadings and placed on three-parameter substrates. The Euler-Lagrange equations are employed to deduce the equations of equilibrium. Then, the asymptotic solutions and boundary value problems are analytically determined by utilizing the two-step perturbation technique. Finally, the effects of the surface parameters, geometric factors, foundation stiffness, volume fraction, thermal effects and layout type of CNTs on the nonlinear bending of the nanobeams are discussed.

Study on nonlinear vibrations of temperature- and size-dependent FG porous arches on elastic foundation using nonlocal strain gradient theory

A nonlocal strain gradient theory is developed in this paper to study the large amplitude vibrations of arches made of functionally graded (FG) porous material. The case of shallow arches resting on nonlinear elastic foundation is modeled via a general higher-order shear deformation theory. The third-order model of Reddy, the first-order model of Timoshenko, and the classical model of Euler–Bernoulli are analyzed. Thermomechanical properties of the arch exposed to the uniform thermal field are assumed to be temperature dependent. The nonlinear motion equations of the arch are established by employing Hamilton’s principle and the von Karman type of geometric nonlinearity. The two-step perturbation technique and the Galerkin method are utilized to solve the nonlinear governing equations. The size-dependent linear and nonlinear frequencies of the arch are obtained for the immovable pinned-pinned boundary conditions. The comparison studies are performed to verify the present solution method with the provided data in the literature, and a good agreement is observed. The novel parametric studies covered in this research include the effects of several parameters such as elastic foundation, nonlocal and length scale parameters, porosity, temperature field, power law index, and geometrical parameters on the frequencies of FG porous arches in detail.

Nonlinear analysis of thermal-mechanical coupling bending of clamped FG porous curved micro-tubes

A nonlinear analysis on the thermal-mechanical coupling bending behavior of functionally graded porous curved micro-tubes is performed in this research. The case of curved micro-tubes with doubly-clamped boundary conditions resting on nonlinear elastic foundation is considered. Based on the modified couple stress theory, curved micro-tubes under uniformly distributed transverse pressure and uniform temperature rise are analyzed. It is assumed that the material properties of the curved micro-tube are temperature-dependent and graded across the radius of cross-section. The governing equations of the curved micro-tube are established within the framework of higher-order shear deformation theory and the von Kármán kinematic assumptions. These equations are reformulated for the case of curved micro-tubes with even distributed porosities based on the uncoupled thermoelasticity theory. The system of nonlinear differential equations governing the equilibrium position of curved micro-tubes is solved using the two-step perturbation technique. The analytical solutions are given in this study to obtain the large deflection in curved micro-tubes as an explicit function of the thermal/mechanical load. The influences of porosity volume fraction, power law index, microstructural length scale parameter, foundation stiffness, and geometrical parameters on the thermomechanical deflection of curved micro-tubes are investigated.

Nonlinear vibration and dynamic instability analyses of laminated doubly curved panels in thermal environments

Nonlinear dynamic analysis for a shear deformable anisotropic laminated doubly curved panel of rectangular planform resting on elastic foundations is presented. A new panel model of arbitrary curvature and arbitrary fiber stacking sequences but constant thickness is developed and established. The governing equations are based on an extended higher order shear deformation shell theory with von Kármán-type of kinematic nonlinearity and including the anisotropic couplings. The nonlinear deformations and initial deflections of the panel are both taken into account. A two-step perturbation technique combining with the Galerkin method is employed to determine the linear and nonlinear vibration frequencies and forced responses. Nonlinear forced harmonic vibration in a Duffing oscillator for anisotropic laminated doubly curved panel is also investigated and discussed together with the effects of damping and excitation by using multiscale and Melnikov methods. The numerical results are validated for various lamination schemes through a comparison with solutions for free vibration frequencies of saddle (hyperbolic), flat plate, cylindrical and spherical shells available in the literature or obtained by the commercial finite element software. The numerical illustrations concern nonlinear vibration and dynamic response of moderately thick, anisotropic laminated doubly curved panels under different values of panel geometric parameters and mechanical excitation.

Nonlinear analysis of thermal-mechanical coupling bending of FGP infinite length cylindrical panels based on PNS and NSGT

The present paper investigates the nonlinear static bending behavior of infinite length cylindrical panels made of functionally graded porous (FGP) materials. The shell with clamped edges is subjected to uniform temperature rise and transverse pressure loading. Thermomechanical properties of the shell with even distributed porosities are temperature-dependent and are graded along the thickness. The principle of virtual displacement and nonlocal strain gradient theory (NSGT) are employed to derive the equilibrium equations of the shallow shell resting on nonlinear elastic foundation. The concept of physical neutral surface (PNS) and higher-order shear deformation theory are also included into the formulation. Using the uncoupled thermoelasticity and Donnell kinematic assumptions, three differential equations of the shell under thermomechanical loading are established. The nonlinear system of governing equations is solved for the shell with infinite length which is clamped on both straight edges and free at other curved edges. The two-step perturbation technique and Galerkin procedure are employed to derive analytical solutions for the thermal, mechanical, and thermomechanical responses of cylindrical panels. The important parameters governing the bending behavior of shells under thermal-mechanical coupling load are identified and discussed. Parametric studies are given to show the influences of nonlocal/length scale parameter, porosity coefficient, foundation stiffness, geometrical parameter and functionally graded pattern. It is shown that the geometrical parameters have an important role on the bending behavior of cylindrical panels. Also, the temperature dependence of material properties results in higher deflection of the heated shells.